Rui Fang， Xiangnan Li， Haixia Wu， Wei Gao and Shiwei Ren,?

（1. Oil Production Technology Research Institute, Dagang Oilfield Company, Tianjin 300280,China；2. School of Information and Electronics, Beijing Institute of Technology,Beijing 100081, China）

Abstract: The concept of difference and sum co-array(DSCA) has become a new design idea for planar sparse arrays. Inspired by the shifting invariance property of DSCA, a specific configuration named here as the improved L-shaped array is proposed. Compared to other traditional 2D sparse array configurations such as 2D nested arrays and hourglass arrays, the proposed configuration has larger central consecutive ranges in its DSCA, thus increasing the DOF. At the same time, the mutual coupling effect is also reduced due to the enlarged spacing between the adjacent sensors. Simulations further demonstrate the superiority of the proposed arrays in terms of detection performance and estimation accuracy.

Key words: planar array geometry；two-dimensional direction of arrival estimation；differencesum co-array；mutual coupling

Direction of arrival(DOA) estimation is a crucial topic in the field of array signal processing, which has useful applications in radar,wireless communication and radio astronomy.Conventional DOA estimation techniques are mainly based on uniformly and regularly placed physical array geometries including uniform linear arrays(ULA) and uniform rectangular arrays(URA)[1-2]. Sensors in these structures are densely positioned, leading to a considerable mutual coupling effect which can cause severe reduction of resolution[3]. Moreover, such structures will also require a higher computational load and hardware cost. Therefore, planar sparse array(PSA) is researched in this paper to recover the signal spatial spectrum with fewer sensors and a sparser physical array configuration.

Inspired by linear sparse array designs such as nested arrays[4]and coprime arrays[5], numerous PSAs have been subsequently proposed for the purposes of 2-dimensional(2D) DOA. Lshaped arrays(LA)[6]and 2D coprime arrays[7]estimate elevation and azimuth angle separately by using their two subarrays and then paring the results. These two configurations successfully solve the ambiguity problem with a fast spectrum searching method, but the number of resolvable sources does not increase significantly.To achieve a hole-free virtual URA with larger degrees of freedom(DOF)[8], several configurations based on the concept of difference coarray(DCA) are proposed including 2D nested array(2D-NA)[9]and open box array[10]. However,dense sensor array sections still exist in these two configurations. To further reduce the mutual coupling effect, improved configurations such as partially open box array(POBA), half open box array(HOBA), half open box array with two layers(HOBA-2) and hourglass array(HA)[10-11]are proposed. Note that the aforementioned array configurations only use the concept of DCA. In fact, sum co-arrays can also be generated by using a certain algorithm. To generate a virtual difference and sum co-array(DSCA), an improved coprime array has been proposed in a linear context. Since both difference and sum co-arrays are considered, mutual coupling between physical sensors are greatly reduced due to the enlarged separation, and the number of DOFs is also increased. Thus, 2D configuration improvement is possible based on similar idea. A DSCA-based 2D coprime array is also proposed[12-13]. However, its DOF is limited due to the holes in the virtual array, so the properties of DSCA are not fully exploited.

In this paper, an improved L-shaped coprime array (ILCA) geometry is proposed based on the DSCA concept. Some sensors of the conventional LA are moved to their centrosymmetry positions to form two coprime subarrays on both positive x and y axis. Based on the shifting incovariance matrix, the large DOF of LA will not be changed, while the sensor separations are enlarged and the mutual coupling effect is reduced.Simulations verify that the proposed ILCA has better performance than 2DNA and HA.

1 Signal Model

With the mutual coupling considered, the signal is modeled as x(k)=CAs(k)+n(k). C is the mutual coupling matrix which indicate the interference between sensor pairs with different separation. According to Ref. [3], C can be denoted as

With pi,pjas the sensor position of the i-th and the j-th sensors. ||||2denotes the distance between the two points. B is set as the mutual coupling distance. Each sensor only exhibits a mutual coupling effect with the sensor whose separation is less than B, while the effect contributed by other sensors can be ignored. Coupling coefficients are c0=1, c1=0.6, c?=c1ejπ(?-1)/4/?,? ＜B.

In order to generate a DSCA to do the 2D DOA estimation, the vectorized conjugate augmented MUSIC algorithm (VCAM) is introduced to the received vector x(t). The first sensor output is denoted as x1(t) and the ith sensor output as xi(t), i=1,2,··· ,N. By collecting NSsnapshots of both x1(t) and xi(t) with delay τ /=0, the sample vectors [x1(1),x1(2),··· ,x1(NS)]and [xi(1+τ),xi(2+τ),··· ,xi(NS+τ)] can be achieved. By calculating the time average function of x1*(t) and xi(t+τ), the following expression can be obtained

Note that the rank of z is one. The subspace-based DOA estimation algorithms can not be applied unless 2D spatial smoothing(SS) algorithm[14]is used to restore the rank. As such a method requires a consecutive array, only the URA part of the generated DSCA should be considered. After the full rank covariance matrix is generated by the SS algorithm, a 2D subspacebased DOA algorithm such as the Unitary ESPRIT algorithm[15-16]can be exploited. Since the DOA algorithm is based on a virtual DSCA, a larger DOF can be obtained with a sparser configuration compared to the conventional physical uniform rectangular array method.

2 Improved L-shaped Coprime Array

2.1 Definition of ILCA

First, the concept of effective difference coarray (EDCA) is given as follows: For a PSA,denoted as P, and its DSCA, denoted as DS, its EDCA (whose size is denoted as E) is the central URA with largest 2D aperture in DS. To design an array geometry which has a larger DOF and weaker mutual coupling effect, the EDCA of the PSA should be as large as possible while the physical configuration should be sparsely positioned. According to this idea, an improved L-shaped coprime array is proposed by forming two coprime subarrays on both the positive x and y axis and then moving the remaining sensors in the conventional LA to the centrosymmetry positions on the negative axis.

Definition 2For any two positive integers Nx≥2 and Ny≥2, the improved L-shaped coprime array(ILCA) is defined as the following PSA whose sensor positions can be denoted as

where for two integers nx, nyand two positive integers m, n, we have

M1and M2are two arbitrary coprime integers.

An ILCA with Nx=15, Ny=15, M1=3 and M2=4 is given in Fig.1. Note that there are two subarrays on both the x and y axis which is formed by moving the sensors from the uniform linear subarrays of the conventional LA. The blue dots in the figure denote physical sensors and the curves denote the moving trail of different sensors. The sensors on positive axes are positioned on set {3 4 6 8 9 12 15} while the sensors on negative axis are located on {–14 –13 –11 –10–7 –5 –2 –1}. The following property holds for the EDCA of ILCAs.

Fig.1 Improved L-shaped coprime array structure with Nx =15, Ny =15 , M1 =3 and M2 =4

Property 1The EDCA of ILCA has the same aperture as the EDCA of its original LA.Suppose there are N sensors in the configuration,with Nxsensors on the x-axis and Nysensors on the y-axis (excluding the origin (0,0)). Therefore,N=Nx+Ny+1 and the size of the EDCA is(2Nx+1)×(2Ny+1).

ProofAccording to the definition, ILCA is generated by moving the sensors in LA. LA is denoted as PLAwith two sensors located arbitrarily at pi=(nxi,nyi) and pj=(nxj,nyj). If piis moved to its centrosymmetry position, then its new sensor location can be denoted as pi′=(-nxi,-nyi). For any two sensors, their DSCA is

Based on the different moving situations, the DSCA of ILCA has the following cases.

① If both sensors are moved, the newly generated DSCA can be denoted as

② If one sensor is moved, while the other one remains stationary, the newly generated DSCA can be denoted as

As a result, the generated DSCA will not be changed if any physical sensor is moved to its centrosymmetry position. Since the DSCA of both the ILCA and LA are the same, then the EDCA of ILCA is also the same as LA, which has the size E=(2Nx+1)×(2Ny+1).

The DSCA of an ILCA with Nx=15 and Ny=15, M1=3 and M2=4 is shown in Fig.2.An EDCA containing (2×15+1)×(2×15+1)=961 virtual sensors is generated in the center of this DSCA. Thus, better performance can be achieved when the 2DDOA algorithm is operated on it.

Fig.2 DSCA of improved L-shaped coprime array with Nx =15 and Ny =15, M1 =3 and M2 =4

2.2 Comparisons with conventional configurations

To measure the DOF of different arrays, the above EDCA of LA and ILCA are compared to other common 2D PSAs. For an HA[11]with parameters Nxand Ny, the number of sensors can be denoted as Nx+2Ny-2 and the size of its EDCA is EHA=(2Nx-1)(2Ny-1). For a 2D-NA[9]with both subarrays as squares, the side length of the dense and sparse subarrays are denoted as Nxand Ny, respectively. The number of sensors for a 2D-NA is calculated by Nx2+Ny2-1 and the size of the EDCA is E2DNA=(2NxNy-1)NxNy. For example, if the total sensor number is set as 49 and the optimal parameters are used for each configuration; for an HA, EHA=1225 with Nx=25 and Ny=13; for a 2D-NA, E2DNA=1225 with Nx=5 and Ny=5. For the ILCA and LA, ELA=EILCA=2401 with Nx=24 and Ny=24. Therefore, ILCA and LA would have the largest EDCA with the same number of physical sensors.

Meanwhile, to further measure the robustness of different array configurations against the mutual coupling effect, the weight function of sensor separation ?=(?x,?y) must be defined as w(?)=|{(pi,pj)|pi,pj∈S,pi-pj=?}|, where|·| denotes the number of elements. For ILCA,w(1,0)＜Nx, w(0,1)＜Nyand w(1,1)=0. For LA with parameters, Nxand Ny, we have w(1,0)=Nx,w(0,1)=Ny, w(-1,1)=1 and w(1,1)=0.

In addition, for an HA with parameters, Nxand Ny, according to the existing conclusion, we have

For the 2DNA with two subarrays as squares, with the parameters are denoted as Nxand Ny, we have

Therefore, the proposed ILCA has less sensor pairs with smaller separations than LA and 2DNA. A lower mutual coupling effect can be achieved according to its sparse configuration.

3 Simulation

In this section, the root mean square error(RMSE) is used to verify the superiority of our proposed ILCA. At the same time, LA, 2D-NA and HA are also simulated for comparison. Here the RMSE is defined as

RMSE results displayed as a function of snapshots for the different array configurations with signal noise ratio SNR=10 dB are shown in Fig.3. As shown, the ILCA acquires the lowest RMSE. The LA performs better than the HA and 2DNA, but a little bit worse than ILCA. It demonstrates that the new structure has the smallest number of sensor pairs located in close proximity. At the same time, the DOF is increased dramatically. RMSE results obtained at different SNRs varying from -10 dB to 25 dB are illustrated in Fig.4. The number of snapshots is 100. As shown, the ILCA and LA can acquire the lower RMSE when the SNR is larger than 0 dB.The reason why LA achieves lower errors when the SNR is less than 10 dB is that the weight functions of w(1,0) and w(0,1) for ILCA are much lower than that of LA. It leads to the reduced coupling effect as well as weak robustness.

Fig.3 RMSE results as a function of snapshots for different array configurations with SNR=10

Fig.4 RMSE results as a function of SNR for different array configurations with snapshots as 100

4 Conclusion

In this paper, an improved L-shaped coprime array based on a difference and sum co-array is proposed. This configuration has the same number of DOFs as the conventional L-shaped array while greatly reducing the mutual coupling effect.Therefore, it can achieve better performance compared to other conventional planar configurations. Simulations results also verified that more accurate estimation results have been achieved by the proposed configurations.